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Chapter 9 High-Order Portfolios

The study of economics does not seem to require any specialised gifts of an unusually high order.

— John Maynard Keynes, Alfred Marshall

Markowitz’s mean–variance portfolio optimizes a trade-off between expected return and risk measured by the variance. However, since financial data is not Gaussian distributed, due to asymmetry and heavy tails in the distribution, it would be reasonable to also incorporate higher-order moments.

Unfortunately, designing a portfolio based on the first four moments (i.e., mean, variance, skewness, and kurtosis) brings at least two critical difficulties:

• The dimensionality of the higher-order moments grows as $N^4$, where $N$ is the number of assets, with implications in the complexity of the moment computation, memory storage, and algorithmic manipulation.

• The portfolio formulations are nonconvex, further complicating the design and optimization.

High-order portfolios were formulated over half a century ago, but have only recently become a practical reality for large numbers of assets in the order of hundreds or even thousands.

This material has been published as: Daniel P. Palomar (2025). Portfolio Optimization: Theory and Application. Cambridge University Press. This version is free to view and download for personal use only; not for re-distribution, re-sale, or use in derivative works. © Daniel P. Palomar 2025.